The LabRI tool is an R Markdown file that employs the indirect method called the LabRI Method. This method is an adaptive and multi-criteria approach for the estimation and verification of reference intervals, utilizing a combination of data cleaning algorithms, data transformation, clustering techniques, and the refineR and reflimR algorithms or the expectation-maximization (EM) algorithm, depending on the number of clusters in the truncated distribution.


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Contents


1. Initial information


Table 1. R Package Status.
AID datawizard ggpubr lattice pacman scales univOutl
TRUE TRUE TRUE TRUE TRUE TRUE TRUE
DT data.table ggtext lubridate plotly shiny xfun
TRUE TRUE TRUE TRUE TRUE TRUE TRUE
FactoMineR devtools grid mclust prettydoc shinyjs writexl
TRUE TRUE TRUE TRUE TRUE TRUE TRUE
KernSmooth digest gt mixR qqplotr shiny.exe zlog
TRUE TRUE TRUE TRUE TRUE TRUE TRUE
MASS dplyr imputeTS modeest readr shinythemes
TRUE TRUE TRUE TRUE TRUE TRUE
MethComp epiR installr moments readxl stats
TRUE TRUE TRUE TRUE TRUE TRUE
RVAideMemoire factoextra irr multimode refineR stringi
TRUE TRUE TRUE TRUE TRUE TRUE
calibrate ffp janitor multiway reflimR systemfonts
TRUE TRUE TRUE TRUE TRUE TRUE
cartography forecast kableExtra nortest reshape2 tools
TRUE TRUE TRUE TRUE TRUE TRUE
cluster ggplot2 knitr openxlsx rmarkdown utf8
TRUE TRUE TRUE TRUE TRUE TRUE


  • Responsible person: Alan Carvalho Dias;


  • Measurement procedure and Method: Measurement Procedure: Beckman Coulter AU 5800 (Brea, California, US); Analytical Method: Gamma-glutamyl-3-carboxy-4-nitroanilide.;


  • Name of the measurand: Amylase;


  • Unit of measurement: U/L;


  • Type of blood specimen: serum;


  • Exclusion criteria: Exclusion criteria included individuals with a body mass index (BMI) >35 kg/m2, consumption of ethanol >= 70 g per day, smoking >20 cigarettes per day, taking regular medication for a chronic disease (diabetes mellitus, hypertension, hyperlipidemia, allergic disorders, depression), recent (< 15 days) recovery from acute illness, injury or surgery requiring hospitalization, carrier of HBV, HCV or HIV, pregnant or within 1 year after delivery. Written informed consent was obtained after written/verbal explanation of the study. Those with any chronic disease were excluded except for individuals aged 50–65 years who had well controlled hypertension taking up to 2 drugs. A single measurement of blood pressure, abdominal circumference and BMI was done after filling the study questionnaire.;


  • Data source: Recruitment of study participants in Kenya was carried out between January and October 2015 in several counties. Majority were urban dwellers from the capital city Nairobi, Kiambu county in central Kenya, Kisii County in western Kenya, and Nakuru County based in the Great Rift Valley. Obtained from the dataset of the study by Omuze et al., available in the Dryad repository: https://doi.org/10.5061/DRYAD.NVX0K6DNS;


  • Age range: 18 to 65 years;


  • Sex: Male and Female;


  • Settings:
    • Number of decimal places: 1;
    • Setting the limits of the Reference Interval (‘Double-sided’ or ‘One-sided’): Double-sided. There is a Lower Limit and an Upper Limit of reference.;


    2. Descriptive statistics


    Table 3. Measures of Position - Part 1.
    Statistical parameters Results
    Sample size (N) 533.0
    Sampling (n) 533.0
    Minimum 27.0
    Mode 84.9
    Mean 91.7
    Median 86.0
    Maximum 327.0
    Table 4. Measures of Position - Part 2.
    Statistical parameters Results
    Sample size (N) 533.0
    Sampling (n) 533.0
    1st percentile 40.6
    2.5th percentile 46.3
    5th percentile 52.0
    10th percentile 56.0
    16th percentile 62.0
    25th percentile 69.0
    50th percentile (median) 86.0
    75th percentile 107.0
    84th percentile 118.0
    90th percentile 131.0
    95th percentile 152.2
    97.5th percentile 172.7
    99th percentile 194.8
    Table 5. Measures of Dispersion.
    Statistical parameters Results
    Sample size (N) 533.0
    Sampling (n) 533.0
    Standard Deviation (SD) 33.2
    Variance 1101.5
    Interquartile Range (IQR) 38.0
    Range 300.0


    3. Estimation Module


    • The LabRI method is an adaptive and multi-criteria approach for the indirect estimation and verification of reference intervals. It integrates data cleaning, data transformation, clustering techniques, and applies the refineR, reflimR, and Expectation-Maximization (EM) algorithms. The method combines parametric and non-parametric percentile approaches to estimate population reference intervals, depending on the number of clusters identified in the truncated distribution.
      Characteristics of the LabRI Method:
    • Adaptive: Adjusts its application based on the structure and characteristics of the data, using different cleaning and transformation techniques as needed. Applies the Centroid of Windsorized Reference Limits method using refineR and reflimR if the data distribution has more than one cluster for reference interval estimation. If there is only one cluster, the expectation-maximization algorithm is used with both parametric and non-parametric approaches to obtain the best reference interval estimate.
    • Multi-criteria: Considers multiple criteria and methods for the estimation and verification of reference intervals, ensuring a robust and comprehensive analysis.


    3.1. Data preprocessing


    Table 6. Outliers removed after iterative cycles of Tukey’s test using the iboxplot() function from the ‘reflimR’ package.
    Statistics parameters Results
    Sample size (N) 533.000
    Sampling (n) 533.000
    Detection of outliers based on Box-Plot 3.000
    Percentage of outliers detected and removed 0.600
    nNew1 530.000
    SkBowleyorig 0.081
    SkBowleytrans -0.004
    |ΔSk| 0.085
    Footnote

    nNew1, Sampling after removal of outliers. SkBowley, Bowley’s Coefficient of Skewness. SkBowleyorig, SkBowley for the original data. SkBowleytrans, SkBowley for the transformed data. |ΔSk|, the absolute difference in the SkBowley values between the original and transformed data.

    Reference: [1], [2], [3], [4]

    3.1.1. Outlier removal and transformation approaches:

    • 1º) To identify and remove outliers in datasets with a sample size of less than 1000, the iboxplot algorithm from the reflimR package is employed. For datasets with a sample size of 1000 or more, the iboxplot is applied as the initial step in data cleaning, followed by the Semi-Interquartile Range (SIQR) boxplot approach to selectively remove upper outliers.
    • 2º) The Box-Cox transformation (see Section 3.1.2) is applied based on the Bowley's Coefficient of Skewness (SkBowley). After the transformation, SkBowley is reassessed to determine which dataset exhibits a distribution closest to Gaussian. The dataset with the most Gaussian-like profile, whether from the cleaned original data or the transformed data, will be used in the clustering and truncation steps.


    3.1.2. Algorithm for data transformation:

    • The ‘Transformation Algorithm’ will be used to select the best Box-Cox transformation (method = log-likelihood) and the optimal lambda value, which is the one that provides the best approximation to a normal distribution. SkBowley is calculated before and after the transformation. To achieve this, a specific treatment is applied to the dataset with adjustments to the distribution tails, aiming to minimize the influence of pathological results on the accuracy of SkBowley values. Lambda (λ) values are truncated between 0 and 3 to limit the transformation to a more stable range, avoiding extreme values that could distort the data distribution. The decision to use the transformed data will depend on the SkBowley values and the difference (delta) between the values before and after the transformation (ΔSk).
    • The transformation of y is performed using eq.(1):

    \[ \Large{ y^{(\lambda)} = \left\{ \begin{matrix} {\frac{y_i^\lambda - 1}{\lambda}}, & \mbox{if } \lambda \neq 0 \\ ln(y_i), & \mbox{if } \lambda = 0 \end{matrix} \right. } \quad \quad \quad \quad \quad \quad {(1)} \]

    • \(y^{(\lambda)}\): the transformed value of the (i)-th observation after applying the Box-Cox transformation.

    • \(y_{i}\): the original value of the \(i\)-th observation in the dataset.

    • \(\lambda\): the transformation parameter that maximizes the normality of the data. It is determined through optimization.

    • \(ln(y_{i})\): The natural logarithm of \(y_{i}\), used when \(\lambda = 0\).


    3.1.3. Interpreting the shape of data distributions:

    • Skewness refers to a distortion or asymmetry that deviates from the symmetrical bell curve, or normal distribution, in a set of data. If the curve is shifted to the left or right, it is said to be skewed.
      • Positive skewed distribution (or right-skewed distribution): It is a type of distribution in which most values are clustered around the left tail of the distribution, while the right tail of the distribution is longer, indicating a positive skewness coefficient (Sk > 0).
      • Negative skewness (or left-skewed distribution): It is a type of distribution in which most of the values are grouped around the right tail of the distribution, while the left tail of the distribution is longer, indicating a negative skewness coefficient (Sk < 0).

    • Kurtosis: Similar to ‘skewness’, kurtosis coefficient (Kt) is a statistical measure used to describe a distribution. While skewness differentiates extreme values in one tail versus the other, kurtosis measures extreme values in both tails. Kurtosis is a measure of the combined weight of the tails of a distribution relative to the center of the distribution.
      • Leptokurtic: A leptokurtic distribution shows heavy tails on both sides, indicating large discrepant values. Theoretically, a leptokurtic distribution is one with K > 3. From a practical standpoint, a leptokurtic distribution with Kt > 3.3 can be considered visibly or perceptibly leptokurtic.
      • Mesokurtic: Data that follows a mesokurtic distribution shows an excess kurtosis of zero or close to zero. This means that if the data follows a normal distribution, it follows a mesokurtic distribution. Theoretically, the kurtosis of a normal distribution is equal to 3. A distribution of results between 2.7 and 3.3 will be considered mesokurtic.
      • Platykurtic: The kurtosis reveals a distribution with flattened tails. The flat tails indicate few outliers in the distribution. Theoretically, a platykurtic distribution is one with kurtosis below 3. From a practical standpoint, a distribution with kurtosis below 2.7 can be considered platykurtic.



    3.1.1. Data shape before cleaning



    • Distribution profile of data before the outlier removal algorithm.


    Table 7. Data distribution before outlier detection and exclusion (n = 533 ).
    Statistical parameters Results
    Sample size (N) 533
    Sampling (n) 533
    Kurtosis coefficient (Kt) 8.73
    Interpretation of the Kt The distribution can be considered leptokurtic.
    Pearson’s coefficient of skewness (SkPearson) 1.61
    Interpreting the result of the SkPearson Empirical relationship between Mean, Median, and Mode is: Mean > Median > Mode. The distribution is highly skewed to the right.
    Footnote
    Mesokurtic: If the Kt is between 2.7 and 3.3, the distribution can be considered mesokurtic.
    Platykurtic: If the Kt is less than 2.7.
    Leptokurtic: If the Kt is greater than 3.3.
    Distribution is highly skewed: If the SkPearson is less than -1 or greater than 1.
    Distribution is moderately skewed: If the SkPearson is between -1 and -0.5 or between 0.5 and 1.
    Distribution is slightly skewed: If the SkPearson is between -0.5 and -0.15 or between 0.15 and 0.5.
    Distribution is approximately symmetrical:
    If the SkPearson is between -0.15 and 0.15.

    References:[5],[6],[7],[8],[9]


    3.1.2. Data shape after cleaning



    • Distribution profile of the data after the outlier removal algorithm.


    Table 8. Data shape after outlier detection (nNew1 = 530 ).
    Statistical parameters ResultS
    Sample size (N) 533
    Sampling (n) 533
    Sample size after removal of outliers (nNew1) 530
    Kurtosis coefficient (Kt) 4.46
    Interpretation of the Kt The distribution can be considered leptokurtic.
    Pearson’s coefficient of skewness (SkPearson) 1.06
    Interpreting the result of the SkPearson Empirical relationship between Mean, Median, and Mode is: Mean > Median > Mode. The distribution is highly skewed to the right.
    Footnote
    Mesokurtic: If the Kt is between 2.7 and 3.3, the distribution can be considered mesokurtic.
    Platykurtic: If the Kt is less than 2.7.
    Leptokurtic: If the Kt is greater than 3.3.
    Distribution is highly skewed: If the SkPearson is less than -1 or greater than 1.
    Distribution is moderately skewed: If the SkPearson is between -1 and -0.5 or between 0.5 and 1.
    Distribution is slightly skewed: If the SkPearson is between -0.5 and -0.15 or between 0.15 and 0.5.
    Distribution is approximately symmetrical:
    If the SkPearson is between -0.15 and 0.15.

    References:[5],[6],[7],[8],[9]


    3.1.3. Box-Cox transformation


  • The magnitude of SkBowley and the of |ΔSk| indicated the need to apply the Box-Cox transformation after data cleaning.


    • Table 9. Shape of the data after transformation (nNew1 = 530 ); Lambda = 0; Method = loglik.
    Statistical parameters ResultS
    Sample size (N) 533
    Sampling (n) 533
    Sample size after removal of outliers (nNew1) 530
    Kurtosis coefficient (Kt) 2.87
    Interpretation of the Kt The distribution can be considered mesokurtic.
    Pearson’s coefficient of skewness (SkPearson) 0.11
    Interpreting the result of the SkPearson Empirical relationship between Mean, Median, and Mode is: Mean > Median > Mode. The distribution is approximately symmetric.
    Footnote
    Mesokurtic: If the Kt is between 2.7 and 3.3, the distribution can be considered mesokurtic.
    Platykurtic: If the Kt is less than 2.7.
    Leptokurtic: If the Kt is greater than 3.3.
    Distribution is highly skewed: If the SkPearson is less than -1 or greater than 1.
    Distribution is moderately skewed: If the SkPearson is between -1 and -0.5 or between 0.5 and 1.
    Distribution is slightly skewed: If the SkPearson is between -0.5 and -0.15 or between 0.15 and 0.5.
    Distribution is approximately symmetrical:
    If the SkPearson is between -0.15 and 0.15.

    References:[5],[6],[7],[8],[9]


    3.2. Clustering and truncation


    3.2.1. Criterion based on the Bayesian Information Criterion (BIC) to identify the number of clusters:





    3.2.2. Criterion based on the number of modes to identify the number of clusters. This criterion uses an algorithm that determines the number of modes based on kernel bandwidth:

    Table 10. Modes and Anti-modes (nNew1 = 530 ).
    Quantitative (n) Modes Anti-modes
    1 4.4201507917 –
    2 – –
    3 – –
    4 – –
    5 – –
    6 – –
    7 – –



    3.2.3. Clustering and truncation of the distribution of results after analyzing the Bayesian information criterion (BIC) and the number of modes:

    Table 11. Statistical parameters estimated by Maximum Likelihood after Box-Cox transformation (nNew1 = 530 ).
    Statistical parameters Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5 Cluster 6 Cluster 7
    Global Midpoint (Mode or Median) 4.4201507917 N/A N/A N/A N/A N/A N/A
    Cluster Mean 4.4578455362 N/A N/A N/A N/A N/A N/A
    Cluster Standard Deviation 0.3260933023 N/A N/A N/A N/A N/A N/A
    Mixture Proportion (Weight) 1 N/A N/A N/A N/A N/A N/A
    Cluster Truncated Distribution (+/- 3SD) 3.3846724782 to 5.5310185942 N/A N/A N/A N/A N/A N/A
    Footnote
    N/A, not applicable.
    The columns with green background indicate the best clusters used to estimate the ‘Truncated Global Distribution’.
    Truncated global distribution (all clusters selected): 3.555420916 to 5.3981791267 .

    Table 12. Shape of the ‘Truncated Global Distribution’ data (nNew2 = 530 ).
    Statistical parameters Results
    Sample size (N) 533
    Sampling (n) 533
    Sample size after removal of outliers (nNew1) 530
    Sample size after distribution truncation (nNew2) 530
    Kurtosis coefficient (Kt) 2.87
    Interpretation of the Kt The distribution can be considered mesokurtic.
    Pearson’s coefficient of skewness (SkPearson) 0.11
    Interpreting the result of the SkPearson Empirical relationship between Mean, Median, and Mode is: Mean > Median > Mode. The distribution is approximately symmetric.
    Footnote
    Mesokurtic: If the Kt is between 2.7 and 3.3, the distribution can be considered mesokurtic.
    Platykurtic: If the Kt is less than 2.7.
    Leptokurtic: If the Kt is greater than 3.3.
    Distribution is highly skewed: If the SkPearson is less than -1 or greater than 1.
    Distribution is moderately skewed: If the SkPearson is between -1 and -0.5 or between 0.5 and 1.
    Distribution is slightly skewed: If the SkPearson is between -0.5 and -0.15 or between 0.15 and 0.5.
    Distribution is approximately symmetrical:
    If the SkPearson is between -0.15 and 0.15.

    References:[5],[6],[7],[8],[9]


    3.2.4. Selection of the approach adopted for the estimation of the reference interval based on the number of clusters in the truncated interval:



    3.2.4.1. Approach applied in a scenario with 1 cluster in the truncated distribution:



    Table 13. Reference Interval estimates using the Expectation-Maximization algorithm and parametric and non-parametric statistical methods to optimize the estimated interval (approach applied in a scenario with 1 cluster in the truncated distribution).
    Algorithm used RIoptimized(parametric,non-parametric)
    EM Algorithm 47 to 163.5
    Footnote
    RI, reference interval. EM, Expectation–Maximization (EM). N/A, not applicable.


    3.2.4.2. Approach applied in a scenario with multiple clusters in the truncated distribution:



    Table 14. Reference Interval estimates with application of Centroid of Windsorized Reference Limits using refineR and refLimR (approach applied in a scenario with multiple clusters in the truncated distribution).
    before Windorization
    after Windorization
    Algorithm used RI before cleaning RI after cleaning RI before cleaning RI after cleaning Centroid RI
    RI refineR 38.9 to 145.5 41.9 to 148.4 42.7 to 153.7 42.7 to 153.7 43.1 to 155.3
    RI reflimR 44.3 to 162.3 44.3 to 162.3 43.5 to 156.9 43.5 to 156.9
    EM Algorithm — Option disabled — Option disabled
    Footnote
    RI, reference interval. EM, Expectation–Maximization (EM).




    3.3. Estimated reference interval


    Table 15. Reference Interval estimated by the LabRI Method of Amylase ( U/L ) , Age: 18 to 65 years , Sex: Male and Female (n = 530 ).
    LabRI Method 95% RI, Two_sided 90% CI of the LRL 90% CI of the URL
    Results 47 to 163.5 44.8 to 49.2 158.5 to 168.5
    Footnote
    RI, reference interval. CI, confidence interval. LRL, lower reference limit. URL, upper reference limit.

    Main references to develop the LabRI method (Estimation Module): [1], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]

    Comparative Reference Interval: 47 to 164 U/L
    Source of the Comparative Reference Interval: Information available in the article published by Omuse et al., cited as: G. Omuse, K. Ichihara, D. Maina, M. Hoffman, E. Kagotho, A. Kanyua, J. Mwangi, C. Wambua, A. Amayo, P. Ojwang, Z. Premji, R. Erasmus. Determination of reference intervals for common chemistry and immunoassay tests for Kenyan adults based on an internationally harmonized protocol and up-to-date statistical methods. PLoS ONE, 15 (2020): e0235234. https://doi.org/10.1371/journal.pone.0235234


    4. Verification Module


    • It is important for laboratories to verify their RIs before applying them for routine clinical care. This requirement applies to reference intervals derived using the indirect approach.


      About the verification module of the LabRI method:
    • A three-level analysis is performed, and after this analysis, the tool concludes whether the compared reference limits are equivalent or not. The first stage is called “First-Level Analysis ~ Statistical Uncertainty”. The second stage is called “Second-Level Analysis ~ Distance Criterion Based on Equivalence Testing”. The third stage is called “Third-Level Analysis ~ Concordance Evaluation”.


    • The first-level analysis evaluates the magnitude of the statistical uncertainty associated with the reference limits. If this statistical uncertainty is acceptable, the second-level analysis is performed, which compares the reference limit estimated by the LabRI method with the comparative reference limit. Therefore, in the second-level analysis, it is assessed, through equivalence testing, whether the differences between the compared reference limits are practically significant.


    • The third-level analysis is only applied if the interpretation of the equivalence test is “Possible Equivalence” or “Probable Equivalence”, as in these cases, the confidence interval of the equivalence test incorporates one of the equivalence limits. In these cases, three approaches are used in the third-level analysis: Fleiss’ Kappa test, Lin’s Concordance Correlation Coefficient, and Flagging Rates.


    4.1. First-level analysis


    The number of values (results or individuals selected) directly affects the accuracy of the calculation of reference limits. The calculation of the Confidence Interval (CI) for each reference limit allows for validation of the number of selected individuals (sample size). It is generally accepted that the 90% CI for each reference limit should be less than 0.2 times the width of the 95% Reference Interval (RI) in question.

    4.1.1. Interpretation of the first-level analysis

    Descrição da imagem

    4.1.2. Results obtained from the first-level analysis

    Table 16. Evaluating the magnitude of the statistical uncertainty associated with the reference limits.
    Statistical Parameters Lower Limit of RI Upper Limit of RI
    WCI 49.2 - 44.8 = 4.4 168.5 - 158.5 = 10
    WRI 163.5 - 47 = 116.5 163.5 - 47 = 116.5
    WCI/WRI ratio 4.4 / 116.5 = 0.038 10 / 116.5 = 0.086
    WCI/WRI ratio 0.038 0.086
    WCI/WRI limit 0.2 0.2
    Footnote
    WCI, Width of the 90% Confidence Interval; WRI,
    Width of the 95% Reference Interval.

    References:[26],[27],[28],[29],[30]



    4.1.3. Conclusion of the first-level analysis


    Table 17. First-level analysis conclusion.
    First-level analysis conclusion

    The width of the 90% confidence interval (CIW) for the lower (LRL) and upper (URL) reference limits is proportionally small relative to the width of the 95% reference interval (RIW), with a ratio of CIW/RIW ≤ 0.20. Therefore, the magnitudes of these statistical uncertainties validate these reference limits. Proceed to the second-level analysis.
    Footnote

    References: [26],[27],[28]


    4.2. Second-level analysis


    4.2.1. Results obtained from the second-level analysis


    4.2.1.1. Permissible difference for the reference limit (pDRR), based on the “Reporting range” (RR) criterion


    • When the RI is narrow, both ratios can be inflated. To address these situations, we have established a pragmatic criterion where the analytical bias reference limits (DRL) must be equal to or greater than 3 times the “Reporting range (RR)” to validate whether the differences between the reference limits are practically relevant (clinically significant).
    • The RR represents the smallest difference between two reported results, that is, the minimum increment in reported analyte concentrations. In laboratory medicine, this incremental value generally ranges between 0.01 and 10. If the number of digits below the decimal point in reporting test results is 2, 1, or 0, RR is 0.01, 0.1, or 1, respectively.


    Table 18. Assessing the difference between the upper reference limits (URL) using the RU based criteria.
    Statistical parameters Results of bias between upper limits
    |DRL| 0.5
    pDRR = 3 x RR 3 x 0.1 = 0.3
    Conclusion
    The difference between the upper reference limits is discernible from a practical standpoint.
    Footnote
    DRL, absolute difference (modulus) between the reference limits. pDRR, value for the permissible difference for the reference limit, based on the Reporting range (RR). RR, represents the ‘reporting range’, defined as the incremental value selected to capture changes in analyte concentration, designed to reflect background noise caused by analytical imprecision and biological variation, capturing only significant changes when a true alteration is likely to occur. For example, if the number of digits below the decimal point in reporting test results is 2, 1, or 0, then the RR is 0.01, 0.1, or 1, respectively.

    References:[31],[32],[33],[34],[35]
    Table 19. Assessing the difference between the lower reference limits (LRL) using the RU based criteria.
    Statistical parameters Results of bias between upper limits
    |DRL| 0
    pDRR = 3 x RR 3 x 0.1 = 0.3
    Conclusion
    The difference between the lower reference limits is not discernible from a practical standpoint.
    Footnote
    DRL, absolute difference (modulus) between the reference limits. pDRR, value for the permissible difference for the reference limit, based on the Reporting range (RR). RR, represents the ‘reporting range’, defined as the incremental value selected to capture changes in analyte concentration, designed to reflect background noise caused by analytical imprecision and biological variation, capturing only significant changes when a true alteration is likely to occur. For example, if the number of digits below the decimal point in reporting test results is 2, 1, or 0, then the RR is 0.01, 0.1, or 1, respectively.

    References:[31],[32],[33],[34],[35]

    4.2.1.2. Defining the permissible difference for the reference limit (pDRL,CVE), based on Haeckel’s approach of estimating the empirical coefficient of variation (CVE) from the reference interval


    The absolute difference \({ (pD_{LRL} = |LRL_{LabRI} - LRL_{Comp}|)}\) between the two lower reference limits \({(LRL_{LabRI} ; LRL_{Comp})}\) compared should not exceed a permissible difference (\({ pD_{LRL}}\)) eq.(2). Similarly, the absolute difference\({ (pD_{URL} = |URL_{LabRI} - URL_{Comp}|)}\) between the two upper reference limits (\({URL_{LabRI}; URL_{Comp}}\)) compared should not exceed a permissible difference (\({ pD_{URL}}\)) eq.(3).


    \[\large{pD_{LRL} \leq \ 1.645 \times pS_{A,LRL} \ }\ \ \ \ \ \ \ \ \ {(2)}\]\[\large{pD_{URL} \leq \ 1.645 \times pS_{A,URL} \ }\ \ \ \ \ \ \ \ \ {(3)}\]

    • \(EI_{LRL}\): Equivalence Interval (EI) of the Lower Reference Limit (LRL).
    • \(EI_{URL}\):Equivalence Interval (EI) of the Upper Reference Limit (URL).
    • \(pS_{a,LRL}\): analytical standard deviation permitted (or Clinically irrelevant uncertainty) at the Lower Limit (LRL).
    • \(pS_{a,URL}\): analytical standard deviation permitted (or Clinically irrelevant uncertainty) at the Upper Limit (URL).
    • \(Factor \ 1.645\): The coverage factor 1.645 is used to calculate a one-tailed confidence interval of 95%.


    Table 20. Permissible difference for the reference limit (pDRL) based on the approach proposed by Haeckel et al., integrating state-of-the-art models with those based on biological variation.
    Method CVE pCVA Medln pSA,Med Slope Intercept pSA,LRL pSA,URL pDLRL pDLRL,CVE pDURL pDURL,CVE
    RILabRI 0.33 0.06 87.66 4.99 0.05 1 3.14 8.44 5.16 5.16 13.88 13.88
    RIComp 0.33 0.06 87.80 5.00 0.05 1 3.14 8.48 5.17 13.94
    Footnote
    RILabRI, reference interval estimated by the LabRI method. RIComp, comparative reference interval. CVE, empirical biological variation coefficient derived from the logarithmic scale, based on the lower reference limit (LRL) and upper reference limit (URL), then converted to the linear scale. If the LRL is not known, it will be set at 15% of the URL for the CVE calculation. pCVA, the permissible CVA at the median on the linear scale is obtained from CVE. Medln, median calculated on the logarithmic scale, as the average of ln(LRI) and ln(LRS), and then converted to the linear scale. pSA,Med, permissible analytical standard deviation at the median of the reference interval. Slope, calculated based on the pSA,Med and at the median of the reference interval. Intercept, is related to the detection limit. Because the detection limit is usually unknown for the matrix of human materials, 20% of the pSA,Med is applied empirically here as a substitute. pSA,LRL, permissible analytical standard deviation at the LRL. pSA,URL, permissible analytical standard deviation at the URL. pDLRL, permissible difference for the lower reference limit. pDLRL,CVE, permissible difference for the LRL, based on Haeckel’s approach estimating the empirical coefficient of variation CVE from the reference interval. pDURL, permissible difference for the URL. pDURL,CVE, permissible difference for the URL, based on Haeckel’s approach estimating the CVE from the reference interval.

    4.2.2. Permissible difference for the reference limit (pDRL) selected as the greater value between pDRU and pDRL,CVE


    Descrição da imagem


    Table 21. Selecting the Permissible Difference for the Reference Limit (pDRL,select).
    Statistical parameters LRL URL
    |DRL| 0.00 0.50
    pDRR 0.30 0.30
    pDRL,CVE 5.16 13.88
    pDRL,select = max(pDRR; pDRL,CVE) 5.16 13.88
    Footnote
    RL, reference limit. RR, Reporting range. DRL, absolute difference (modulus) between the reference limits. pDRR, value for the permissible difference for the RL, based on the RR. pDRL,CVE, permissible difference for the RL, based on Haeckel’s approach estimating the empirical coefficient of variation (CVE) from the reference interval. pDRL,select, permissible difference for the RL, selected as the greater value between pDRR and pDRL,CVE. URL, upper reference limit. LRL, lower reference limit.


    4.2.3. Interpretation of the second-level analysis

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    4.2.4. Conclusion of the second-level analysis


    Table 22. Second-level analysis conclusion.
    Reference limits compared Second-level analysis conclusion
    URLLabRI vs URLComp
    Equivalent. The differences between the compared reference limits are less than or equal to the selected permissible difference (pDURL,select). This pDURL,select is based on the Haeckel approach, which takes into account the empirical coefficient of variation, and the approach that considers the Reporting range (RR).
    LRLLabRI vs LRLComp
    Equivalent. The differences between the compared reference limits are less than or equal to the selected permissible difference (pDLRL,select). This pDLRL,select is based on the Haeckel approach, which takes into account the empirical coefficient of variation, and the approach that considers the Reporting range (RR).


    4.3. Third-Level Analysis


    4.3.1.Interpretation of the third-level analysis


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    4.3.2. Results obtained from the third-level analysis


    Table 24. 3×3 Contingency table comparing value interpretations based on reference intervals (Rows = Comparative Reference; Columns = LabRI method).
    High Low Normal
    High 17 0 0
    Low 0 13 0
    Normal 0 0 500
    Footnote
    Concordance 100.00%
    Low, values below the lower reference limit (LRL). “Normal”, The terms ‘healthy’ or ‘normal’ should not be used anymore because health and normality are relative conditions lacking a universal definition. There is often a gradual transition from ‘health’ to disease. In the present review, the term pathological values are used for values from diseased subjects, and non-pathological values for values from non-diseased subjects. However, for simplicity in this table, the term ‘normal’ is used to represent non-pathological values. High, values above the upper reference limit (URL).
    Table 25. Fleiss kappa statistics for attribute agreement analysis between LabRI method and Comparative Reference in laboratory test Classifications.
    Statistical parameters Low ‘Normal’ High Global
    Fleiss’ kappa 1 1 1 1
    p-value <0.001 <0.001 <0.001 <0.001
    Interpretation Almost perfect agreement Almost perfect agreement Almost perfect agreement Almost perfect agreement
    Footnote

    The interpretation of the kappa values is as follows:
    < 0: Agreement worse than chance. ≥ 0 to ≤ 0.40: Poor agreement. > 0.40 to ≤ 0.60: Weak agreement. > 0.60 to ≤ 0.80: Moderate agreement. > 0.80 to ≤ 0.90: Strong agreement. > 0.90 to ≤ 0.95: Very strong agreement. > 0.95 to ≤ 1.00: Almost perfect agreement.

    The cells are colored based on the kappa value: ‘Very Strong’ to ‘Almost perfect’ agreement. Strong agreement. From ‘Agreement worse than chance’ to Moderate agreement.

    References: [61], [62], [63], [64]
    Table 26. Analysis of the Agreement of Standardized Results (zlog) based on Lin’s Concordance Correlation Coefficient (ρc).
    Statistical parameters Results
    Sample size (N) 533
    sampling (n) 533
    Sample size after removal of outliers (nNew1) 530
    Sample size after distribution truncation (nNew2) 530
    Lin’s Concordance Correlation Coefficient (ρc) 1
    99.5% Confidence interval of the ρc 1 to 1
    Pearson correlation coefficient (ρ) - measure of precision 1
    Bias correction factor (Cb) - measure of trueness 1
    Interpretation Almost perfect agreement
    Footnote

    Interpretation based on the lower limit of the confidence interval for ρc is as follows:
    < 0.90: Poor agreement. ≥ 0.90 to ≤ 0.92: Weak agreement. ≥ 0.92 to ≤ 0.95: Moderate agreement. ≥ 0.95 to ≤ 0.97: Strong agreement. ≥ 0.97 to ≤ 0.99: Very strong agreement. > 0.99: Almost perfect agreement.

    The cells are colored based on the lower limit of the confidence interval for ρc is as follows: ‘Very Strong’ to ‘Almost perfect’ agreement. Strong agreement. Poor to Moderate agreement.

    References: [63], [64], [65], [66]
      Verification of flagging rates:
      • POTD, Percentage Outside Truncated Data. DPORI, Difference in Percentages Out of Reference Interval. PORI, Percentage Out of Reference Interval. PAURL, Percentage Above Upper Reference Limit. PBLLR, Percentage Below Lower Reference Limit. %Y, percentage of result below the truncated range. %Z, percentage of results above the truncated range.
      • The expected flagging rates will be compared with their current rates derived from the calculations of the original indirect study. When the increase in flagging in any direction does not exceed the predefined quality targets, the RI under evaluation can be considered acceptable for use. The goal is to assess the flagging rates to determine whether a change in the historical Reference Interval will create higher flagging rates.
      • Ideally, the percentages of results below the lower reference limit (PBLRL) and above the upper reference limit (PAURL) should be kept reasonably close to 2.5% to avoid significant changes in the sensitivity and specificity of the laboratory test. Based on the principle of the minimum category used to define the allowed bias limits, the flagging rates can reach a value of 5.7% or lower for one of the reference limits, while the other limit would have a value of 1% or lower.
      • Verify if POTD, DPORI, PORI, PAURL, and PBLLR meet the pre-defined criteria based on studies of biological variation. Investigate the adopted partitioning criteria (e.g., age, sex, etc.) Verify the criteria for data exclusion and filtering. If it is identified that any of these parameters do not meet the pre-defined criteria, consider refining the criteria for data exclusion and filtering to achieve better standardization in future studies.
    Table 27. Evaluation of the ‘Flagging Rates’ of Data from the ‘New Truncated Global Distribution’.
    Method Sample size POTD PRWRI PBLRL PAURL DPORI PORI
    RILabRI 530 0.56% 94.34% 2.45% 3.21% 0.75% 5.66%
    RIComp 530 0.56% 94.34% 2.45% 3.21% 0.75% 5.66%
    Footnote

    RILabRI, reference interval estimated by the LabRI method. RIComp, comparative reference interval. POTD, Percentage Outside Truncated Data. PRWRI, Percentage Results Within the Reference Interval. PBLRL, Percentage Below Lower Reference Limit. PAURL, Percentage Above Upper Reference Limit. DPORI, Difference in Percentages Out of Reference Interval. PORI, Percentage Out of Reference Interval.

    When the total percentages of values outside the truncated range, PBLRL, PAURL, DPORI, and PORI,exceed 10%, 5.7%, 5.7%, 4.7%, and 6.7%, respectively, the table cell will have a red background. In these cases, it is recommended to review the exclusion criteria for reference individuals (if direct sampling is used) or the filtering criteria applied to the original database (if indirect sampling is used), or to review the partitioning used by sex or age group.

    References: [8], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76]



    4.3.3. Conclusion of the third-level analysis


    Table 28. Third-level analysis conclusion.
    Reference limits compared Third-level analysis conclusion
    URLLabRI vs URLComp
    The upper reference limits (URLs) were considered equivalent in the second-level analysis. Therefore, the third-level analysis will not be performed.
    LRLLabRI vs LRLComp
    The lower reference limits (LRLs) were considered equivalent in the second-level analysis. Therefore, the third-level analysis will not be performed.
    Flagging Rates - RILabRI
    The Flagging Rates associated with the reference interval estimated by the LabRI method meet the minimum targets based on biological variation studies.
    Flagging Rates - RIComp
    The Flagging Rates associated with the reference interval defined as the Comparative Reference meet the minimum targets based on biological variation studies.